Recently I was reminded of an activity that I used with students that I called The Traffic Jam Activity. View the URLs below if you’re interested in more of the details. There were several ideas that I was working on, but this morning I’m thinking again about concrete (in this case, the little plastic people) manipulatives vs. the virtual (in this case, the Java applet by Mike Morton) manipulatives.

I’m convinced after working for several years with students in a lab environment that using both is better than using one over another. I remember that when I first tried it using both helped me extend the students’ problem solving experience. My students didn’t complain that we were “still” working on a problem or that we were doing a problem “again” if the second time they were trying it with virtual manipulatives. I’m not sure if they considered it to be a different problem or if they liked working on the computer so much they didn’t want to complain or maybe a little of both?

I often observed that for many of my students we had to really work for them to connect the concrete and the virtual experiences. And, in fact, it reminded me of how much we had to work to go from a manipulative environment to a more symbolic environment.

Do you use both kinds of manipulatives when you’re working on a problem? Do you have students have concrete Activity Pattern Blocks (for example) as your students have that applet in front of them? or Tangrams? or Algebra Tiles? or Dice?

I wonder if the conversations that you might have with students as they make connections between the concrete and the virtual might be a starting point for the conversations that you might have with them about going from the concrete to the abstract?

What are your thoughts?

**Some “Traffic Jam” links** in case you are interested:

- ON-Math Fall 2002, Volume 1, Number 1: Developing Algebraic Thinking [PDF version]
- Post-lesson Interview with questions posed to me by Ihor Charischak
- Suzanne’s Lessons: Traffic Jam Activity
- Technology Problems of the Week (tPoWs): Traffic Jam

Teaching solving equations for a variable is a challenge in 6th grade. Students are at a variety of thinking levels from concrete to the abstract. I teach with concrete manipulatives and drawing balance scales on the board to match equations. The biggest “aha” leap my students have made with learning about solving for a variable has been with NLVM Algebra Balance Scales.

http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html

This is a virtual manipulative I use on the smartboard. It seems to be concrete (students can touch and drag), symbolic (equation is what is being manipulated) and abstract in understanding what is happening. My students want to do this endlessly. I think because they can see the connection between the concrete and the abstract.

We have discussions about how the virtual manipulative represents what we are doing mathematically on paper as we solve each equation. Students try to solve the equation first on paper and then take turns solving it step by step on the smartboard. It is a simple but powerful manipulative. Our discussions help those less abstract thinkers reach what we are doing. Interestingly, the struggling students want to try and are not afraid to make a mistake on the smartboard. I think because they can experiment and “undo” when they see the balance scale is going the wrong way. At the same time the class is disucssing what is working and not working and why. It is powerful. Thanks for reminding me to start my year with this fun activity! ;)

This part is very interesting to me:

“I often observed that for many of my students we had to really work for them to connect the concrete and the virtual experiences. And, in fact, it reminded me of how much we had to work to go from a manipulative environment to a more symbolic environment.”

As I work to conceptualize the body’s role in math learning I keep thinking that the first step in accepting this ‘new’ (not!) approach to learning math is in coming to understand how to bring a math idea into many different settings as possible for students. Dienes and Lesh have a lot to say on it. If you take a look at the Lesh translation model (http://www.cehd.umn.edu/ci/rationalnumberproject/03_1.html) can you imagine ‘body-scale’ as a sixth mode? So…we’d have:

Hand-based (manipulatives)

Visual (pictures)

Verbal

Symbolic

In context (‘real life situations’)

Body-based (distruption of scale)

It makes sense to me that each one of those modes provides a slightly different perspective on whatever math idea is under examination. This is the process of abstraction, right?

I wonder if how you’re thinking of the “body-based” mode might not also be considered the “real life situation”.

Interestingly I first encountered the Traffic Jam problem in the textbook I was using with my 7th grade students. In that book (Interactive Mathematics by Glencoe) on page 5 it was called Hop, Skip, Jump. It was an activity designed as an “ice breaker” when you first create groups. Sadly in that mathematics text there was no mention of the mathematics behind the problem at all! It was more like a fun “game.” And even more sad was that it was an absolute disaster the first time I tried it in the 1995-96 school year when I noticed it in the book. It wasn’t until the following year (having coincidently experienced it at a Math Forum Summer Institute: http://mathforum.org/workshops/sum96/traffic.jam.html ) when I had thought about the mathematics behind the “real life situation” that I could find a workable way to introduce the activity at the beginning of my 1996-97 students. By then I had written a Lesson Plan [http://mathforum.org/alejandre/frisbie/jam.html].

I was prepared for the chaos that would probably happen when having students try the “large movement experience” aka “body-based” aka “real life situation.” I didn’t dwell on it but had the students experience it and then went on with the other modes.

As an aside — any of these types of “lessons” I didn’t do continuously or, in other words, I did start a class and continue to the end in one or two or three class periods. This particular activity we spread over a month’s time. One day near the end of class I introduced the “large movement experience.” A day or two or more passed and I introduced the manipulative idea. Again a few days might pass and I introduced the virtual manipulative … and so on. I broke it up over time.

So, after about a month with the different parts of the activity completed, the culminating challenge was that one day I told the students that I would randomly put them in teams of 7. (I had 35 students!) This would be happening the next day and they would have 15 minutes to prepare with their team members. One person would be the “director” and the other six would be the acting out the Traffic Jam. I would give them seven pieces of paper to use as the “stones.” Once the 15 minutes of practice time was up, I would randomly choose the first of the five teams to perform and it would be videotaped.

What was absolutely amazing (considering the chaos of the first large movement experience) is that all 5 teams could perform the Traffic Jam perfectly! My students felt triumphant.

What made the difference?

Were my students able to perform the large movement aka body-based mode because of their work with the concrete manipulatives, virtual manipulatives, verbal and symbolic explanations? Was one a trigger where another wasn’t? Was it the passage of time that helped?

Okay, first thing — what do you think made the difference? Could it have possibly been the fact that you gave them experiences in all those different modes?

I’d love to hear your thoughts about what constitutes ‘real life math’. My current understanding of that phrase would be something akin to modeling mathematical questions that might have a basis in our daily lives but not necessarily experiencing it ourselves w/in the context of the math lesson.

To me, body-scale or whole-body movement in math class can be ‘real life’ b/c we are using our bodies, but there are a couple other things at play here that distinguish it from all the other modes listed in my earlier comment.

First, there is a lot of research happening around body-scale math learning. Essentially, this means taking a math idea/concept that might be worked at hand-scale with manipulatives or on paper or in symbolic form, and doing it at body-scale. For example, in an investigation into polygons you might bring students outside to work in teams with lengths of rope to create the polygons in a new mode. Even our understanding of shapes we think we know on the page/visually can be deepened by changing our perspective and relationship to the ‘known’ shape. The Traffic Jam problem seems to be a great example of this — you can play the game on paper or with little manipulated game-pieces. Or, students’ bodies can become the game pieces themselves, providing new perspectives/insights. I’ve also seen a lot of different versions of “walk the number line” activities. This too is an example of body scale math.

In other words, body-scale math is anything where the whole body is engaged in a mathematical investigation AND the lesson resembles the mathematics at the more standard page level. If you’re using the whole body and it looks like math, then I’m calling it body-scale or whole-body math learning.

[And, this is in juxtaposition to learning math within a dance system, which is a whole other thing.]

Second, I have been reading about embodied cognition in mathematics learning. Overall, it appears that both the theory and research in this discipline support the idea that the body is the source of conceptual understanding of mathematics. (Some would even say the body is the source of mathematics itself.) These new understandings about how the body is an active player in math learning will hopefully help us understand what to look for as our students express themselves mathematically (often through gesture paired with speech) or by providing body-scale modes of learning and developing our abilities to observe how students are thinking about the math ideas in question as they move through body-based tasks.

So, while body-scale math learning *might* be considered “real life” 1) I’m doubtful that Lesh & Dienes were thinking of it in the ways I’ve described above and 2) not all ‘real-life’ math translates into meaningful whole-body movement experiences.

I hope this makes some sense! Please let me know if I can clarify anything. I am very much enjoying hearing about your experience (and thoughts on) using the body in math learning!

What a neat conversation! Some thoughts…

There is lot of research that multiple representations (modes) propel problem solving and conceptual understanding if they are linked within the same problem or context. The act of “translating” between modes makes us smarter.

My definition of “real life” is purely social: something is real life if a community finds utility or beauty in it. For example, a lot of Leonardo da Vinci ideas, like helicopters, only became “real life” hundreds of years later… During his time, these ideas were abstract play. It can be very frustrating that inventions require community adoption to achieve reality!

This spring, we started working on a Wikipedia article, about embodied design. I want to include body-scale math and disruption of scale ideas from this conversation… http://en.wikipedia.org/wiki/Embodied_design_(mathematics_education)

I think what really made the difference was both “time on task” and the “experiences in different modes.” Interestingly I don’t really think that in this case the body-scale or whole-body movement was one of the modes that made a difference but I can see how it could be. In this particular month-long activity the other experiences (concrete manipulative, virtual manipulative, symbolic representation all mixed with conversations and time gaps) were what led to understanding that they demonstrated with the whole-body movement activity.

On further reflection, I agree with you that “real life math” is quite different from “whole-body movement.” I think the reason that I first suggested that whole-body movement might fit into that mode type is because I don’t think there’s always a way to present a problem using “whole-body movement” and I guess I was thinking that it’s too small a subset of problems. As I think about it more, though, just because we offer different modes to approach problems it doesn’t mean that all problems should have to be open to using all modes. After all, many problems are never presented with a real-life context and others don’t lend themselves to manipulatives (concrete or virtual) and … I could go on. So, why not have “whole-body movement” as one additional mode added to the list that you started. Notice that I’ve also added “virtual manipulatives” to your list making 7:

concrete manipulatives [hand-based manipulatives]

virtual manipulatives

visual (pictures)

verbal

symbolic

in context (real life situations)

whole-body movement

I started thinking about some other examples of activities either I’ve been involved with or that I’ve described as having done with my students that I believe involve whole-body movement, including:

Leonardo da Vinci Activity

http://mathforum.org/alejandre/frisbie/math/leonardo.html

Human Graph Theory

http://mathforum.org/workshops/sum98/graph.activity.html

Mathematical Golf

http://mathforum.org/workshops/sum96/photo.album/golf.html

Quadrilateral Activity

http://mathforum.org/alejandre/frisbie/quad.html

And while I didn’t write this in the context of moving, I know that you can imagine the tape on the floor and having students “act it out.”

Integer Operations

http://mathforum.org/dr.math/faq/faq.integer.alej.html